How Do You Solve an Epidemic Model Using Partial Fractions?

[ISU20CpreE]

Partial Fractions:
A single infected individual enters a comunnity of n susceptible individuals. Let x be the number of newly infected individuals at time t. The common epidemic model assumes that the disease spreads at a rate proportional to the product of the total number infected and the number not yet infected.So
$$\frac {dx} {dt} = k(x+1) (n-x)$$ and you obtain $$\int\frac {1} {(x+1)(n-x)} dx = \int k dt$$ I need to know how to set up the problem and then work from there.

Any suggestions.

 

[Tide]

You have already "set up" the problem.

I think you're looking for this:

$$\frac {1}{(x+1)(n-x)} = \frac {1}{n+1} \left( \frac {1}{x+1} + \frac {1}{n-x}\right)$$